OS X Yosemite installation error: File system verify or repair failed

You could try booting off a Linux LiveUSB to back up your files.

Once that is done, you can reformat your HD and then reinstall OS X.

You should also keep an eye on your HD; these types of errors often suggest that your drive is corrupted or damaged.

Have you got access to another mac? Your best bet would be to connect other mac using target disk mode, either via firewire or thunderbolt. Your internal drive will show up on the desktop of the other mac for you to copy off your files.

Other option would be booting up via an other system via an external drive, open disk utility and try to repair a couple of times. Soon or later your internal drive will pop up on the desktop so you can recover your files. But I stress I need an external hard drive or USB key drive with a OS X system on it to do this.

It's sounds like your hard drive is dying basically, hence why Disk Utility can fix/repair for you to install Yosemite.

What mac are you running?

1. Attach your thumb-drive and copy all contained files & folders to your external drive, then start Disk Utility.
2. Copy Install OS X Yosemite to your external drive
3. Choose your thumb drive in the left pane and repartition it to 1 Partition, click the Options-button, choose GUID Partition Table and OK, then Apply

4. Quit Disk Utility and open Install OS X Yosemite
5. Install Yosemite to your thumb drive. You may have to click Show all partitions if your thumb-drive isn't visible initially.
6. After a automated restart configure Yosemite. You will be booted to the thumb drive now.
7. Start Terminal in Applications/Utiilities/

8. At the prompt enter df and hit ↵

   The output looks similar to that:

   Filesystem    512-blocks       Used  Available Capacity   iused    ifree %iused  Mounted on  
   /dev/disk1s2   121872696   67379640   53981056    56%   8486453  6747632   56%   /
   devfs                387        387          0   100%       670        0  100%   /dev
   /dev/disk0s2   546875000  328125000  217547952    60% 735912806 22434744   60%   /Volumes/SystemInternalDrive
   map -hosts             0          0          0   100%         0        0  100%   /net
   map auto_home          0          0          0   100%         0        0  100%   /home
   /dev/disk2s2  2923093280 1922781536 1002781536    66% 487048844  1269484   66%   /Volumes/ExternalDrive
   /dev/disk2s3   733093280  142781536  593111744    20% 487048844  1269484   20%   /Volumes/SystemExternalDrive
   

9. At the prompt enter diskutil list and hit ↵

   The output looks similar to that:

   /dev/disk0
      #:                       TYPE NAME                    SIZE       IDENTIFIER
      0:      GUID_partition_scheme                        *400.2 GB   disk0
      1:                        EFI                         209.7 MB   disk0s1
      2:                  Apple_HFS SystemInternalDrive     280.0 GB   disk0s2
      3:                 Apple_Boot Recovery HD             650.0 MB   disk0s3
      4:                     MS-DOS WinSystem               120.0 GB   disk0s4
   /dev/disk1
      #:                       TYPE NAME                    SIZE       IDENTIFIER
      0:      GUID_partition_scheme                        *62.7 GB    disk1
      1:                        EFI EFI                     209.7 MB   disk1s1
      2:                  Apple_HFS SystemUSB               62.4 GB    disk1s2            
   /dev/disk2
      #:                       TYPE NAME                    SIZE       IDENTIFIER
      0:      GUID_partition_scheme                        *2.0 TB     disk2
      1:                        EFI                        209.7 MB    disk2s1
      2:                  Apple_HFS ExternalDrive           1,5 TB     disk2s2
      3:                  Apple_HFS SystemExternalDrive    500.0 GB    disk2s3    
   

   where SystemInternalDrive is the name of your internal system partition,
   WinSystem is the name of your internal bootcamp partition,
   SystemUSB is the name of your thumb drive system partition,
   ExternalDrive is the name of your external data partition and
   SystemExternalDrive is the name of your external system partition (if you have one)

   If your output looks really different, particularly if you have a Apple_CoreStorage device stop here. You have to post the output here and i have to adapt my answer

   Please replace all names in the commands below with your original partition names.

10. Data copying: You need at least 280 GB free space on your external hard disk to copy your Mac partition. You need additional free space there in the size of your Windows partition if you want to copy that also. Before starting the copy open Disk Utility and unmount the partition SystemInternalDrive and possibly WinSystem then quit Disk Utility.

    Now enter in Terminal.app:

      sudo dd bs=512 if="/dev/rdisk0s2" of="/Volumes/ExternalDrive/RescueMac.dmg" conv=noerror,sync
    

    hit ↵, enter your password (it will not be visible) and hit ↵ again to copy your internal Mac volume to an image file on your external disk though it contains errors. The input file (if) is the Device Indentifier of the corrupted partition found in the diskutil list output above prepended by /dev/r hence /dev/rdisk0s2.
    This will take a good while. If you enter a higher value for bs it should be faster. Using 512 i got 4 MB/s, using 524288 (1024x512) i got about 65 MB/s (but wasn't successful copying my thumb-drive finally). Often a larger integral multiple of 512 is recommended (e.g. 4096 bytes=(8x512)).

    If you also want to copy your boot camp partition that might work:

      sudo dd bs=512 if="/dev/rdisk0s4" of="/Volumes/ExternalDrive/RescueWindows.dmg" conv=noerror,sync
    

11. After the process is finished enter exit and hit ↵, quit Terminal, open your external disk with the Finder and open the newly created RescueMac.dmg file. Check if you can open some files or folders.

12. Start Disk Utility and choose your internal disk in the left pane
13. Choose Erase and click the Security Option…-button, choose single pass of zeroes like in the picture below and OK then Erase (In the pictures below i have chosen to erase my thumb drive. You shouldn't do that. Since i started from my internal disk – named VMWare Virtual->System – i wasn't able to demonstrate the erasure with that one because it's the start volume and everything is greyed out) The zeroing will map out bad blocks.
14. Open Install OS X Yosemite on your external drive and install it to your newly created internal drive. You may have to click Show all partitions if your internal drive isn't visible initially.
15. After a automated restart configure Yosemite. You will be booted to your internal drive now.
16. Open RescueMac.dmg on your external drive and copy all needed files to the respective folders in your user folder. Some of your files may be corrupted because they resided on your defective hard disk and got zeroed out in the earlier executed dd task
17. Get a new internal hard disk as soon as possible and replace your probably corrupted old one like octosquidopus mentioned in his answer